Error control of a numerical formula for the Fourier transform by Ooura's continuous Euler transform and fractional FFT

TL;DR

This paper presents an improved numerical method for Fourier transform computation of slowly decaying functions, combining Ooura's continuous Euler transform with fractional FFT to achieve accurate results efficiently across specified frequency ranges.

Contribution

It introduces a parameter selection strategy based on error analysis for Ooura's formulas and integrates fractional FFT to enhance computational efficiency.

Findings

Achieves accurate Fourier transform computation within specified frequency ranges.

Reduces computational time to that of standard FFT.

Demonstrates effectiveness of combined method through numerical experiments.

Abstract

In this paper, we consider a method for fast numerical computation of the Fourier transform of a slowly decaying function with given accuracy in given ranges of the frequency. In these decades, some useful formulas for the Fourier transform are proposed to recover difficulty of the computation due to the slow decay and the oscillation of the integrand. In particular, Ooura proposed formulas with continuous Euler transformation and showed their effectiveness. It is, however, also reported that errors of them become large outside some ranges of the frequency. Then, for an illustrating representative of the formulas, we choose parameters in the formula based on its error analysis to compute the Fourier transform with given accuracy in given ranges of the frequency. Furthermore, combining the formula and fractional FFT, a generalization of the fast Fourier transform (FFT), we execute the computation in the same order of computation time as the FFT.